Research and application of compact finite difference method of low reaction diffusion equation

Journal of Chemical and Pharmaceutical Research, 2014, 6(3):27-33

Research Article

_{CODEN(USA) : JCPRC5}

ISSN : 0975-7384

Research and application of compact finite difference method of low

reaction-diffusion equation

Huancheng Zhang

1

_{, Guanchen Zhou}

1

_{, Yingna Zhao}

2

_{and Junpeng Yu}

1

1

_{Qinggong College, Hebei United University, Tangshan, China}

2

_{Hebei United University, Tangshan, China}

_____________________________________________________________________________________________

ABSTRACT

The paper describes the theory of fractional derivative and specific application examples in the field of engineering sciences. On this basis, this paper mainly studies the low reaction-diffusion equations. First using compact operator, the paper constructs a higher-order finite difference scheme. Then the paper proves the existence and uniqueness of the difference solution by matrix method and analyzes the stability and convergence of the scheme by Fourier method.

Key words:Fractional Differential Equations, Low reaction-diffusion equations, Compact finite difference method, Fourier method

_____________________________________________________________________________________________

INTRODUCTION

Application of fractional differential equations in the field of engineering sciences is gradually expanded. Especially in recent years, application of fractional differential equations in the field of hydromechanics, viscoelasticity, rheology, fractional control systems and fractional controller, electroanalytical chemistry, electronic circuit and electrically conductive in biological system are more and more [1-3].

There are three fractional derivative definitions: Griinwald-letnikov (G-L) Definition[1], Riemann-Liouville (R-L) Definition[1] and Caputo Definition[2].

R-L Definition:

Let (0,1),a,bR,atb,f(t) is continuous on [a,b]_{, the R-L fractional differential is}  



 

 _d

t f dt d t

f t

a



 

 

 1

1 t a

) (

) ( ) ( 1 ) ( D

where



(



)

is Gamma function.

Caputo Definition:

.

1

0

,

0

,

)

(

)

(

)

1

(

1

)

(

a





























_f

_t

_f

_s

_t

_s

_ds

_t

_T

D

t

a t

When



negative real number or positive integer, three definitions is can be converted to each other. G-L definition is generally used for discrete computing. R-L and Caputo definition are commonly used in the discussion of fractional differential equations.

APPLICATION EXAMPLE

(1) Promotion and application of Newton's law

(2) Conduction applications in biology

In the study of biological electrical conduction, experts give the transfer function ( ) 0 ,(0 1)



_





_X 

X

where



is current frequency,

X

0 _and



_{are constant and their values are ​ ​ associated with the cell type.}

If we see the above formula as Laplace transform, i.e. S G(s)G0



, then L inverse transformation is fractional

differential equation 0

D

l

g

(

t

)



0

,

0







1

_.

(3) Application of

PI



D

controller

The function of

PI



D

 controller is

  _{k s} s k k s E

s U s

G   _p _I   _D

) (

) ( ) (

where



,



0,kp,kl,kDconstants are and

the output equation is klD e(t)kDD e(t)kpe(t)u(t)

 

.

For the above formula, if









1

, then it is the traditional PID controller; if





1

,





0

, then it is PI controller; if





0

,





1

, then it is PD controller; if









0

, we give a gain.

(4) Application of Fractional control system The system equation of

PI



D

 controller is

 

(

)

(

)

(

)

(

)

(

)

)

(

0

t

w

D

k

t

w

D

k

t

w

k

t

y

D

k

t

y

D

k

t

y

k

t

y

D

a

p I D p I D

n

k

k k    



_

_



_

_

_



_





Where the transfer function is





 















_m

k

D I p k

D I p closed

s

k

k

s

k

s

a

s

k

k

s

k

s

G

k

0

)

(

  

 

  

.

If the system is open-loop system, then the differential equation is

) ( )

( )

( ) (

0

t w D k t w D k t w k t y D

a p I D

n

k

k k

 

 _{ }  _





.

COMPACT FINITE DIFFERENCE METHOD OF LOW REACTION-DIFFUSION EQUATION Derivation of the equation

General reaction-diffusion equation set is the following:

)

,

(

)

,

(

)

,

(

)

,

(

₂

2

t

x

b

t

x

a

t

x

a

x

D

t

x

a

t















(1)

) , ( ) , ( ) , ( )

,

( ₂

2

t x b t x a t x b x D t x b

t  



  



(2)

Where D is diffusion constant. When the particle movement and reactions are affected by the low diffusion factor, equation set can be developed in the following form

























₍

_,

₎



₍

_,

₎

₍

_,

₎

₍

_,

₎

2 2 1

0

a

x

t

a

x

t

b

x

t

x

D

t

x

a

t

t









(3)

 

_

  

 

 

 



 

) , ( ) , ( , )

,

( _{2 2}

2 1

0 bx t a x t b x t

x D

t x b

t t









(4)

where



 is diffusion coefficient, 0

D

t1



(

x

,

t

)

















_

 _d

t x t t

x D

t

t







 

  

0 1 1

0 _{( )}

) , ( )

( 1 ) , (

(5)

By decoupling operation, Chen, Liu and Burrage[3] simplified the formula (3) and (4) for the following low-reaction-diffusion equation:

L

x

T

t

t

x

f

t

x

u

t

x

u

x

D

t

x

u

t

t

_























0

,

0

),

,

(

)]

,

(

)

,

(

[

)

,

(

₂

2 1

0 









(6)

where 0 1,

k





0

_{is general diffusion coefficient, k}_{0 is bimolecular reaction rate constant.}

Dirichlet boundary and initial conditions of (6) are

T

t

t

t

u

(

0

,

)





(

),

0





₍₇₎

T

t

t

t

L

u

(

,

)





(

),

0





₍₈₎

L

x

x

x

u

(

,

0

)





(

),

0





₍₉₎

For the initial boundary value problem of this equation, Chen et al[3] gave implicit difference scheme and explicit difference scheme. They respectively proved the stability of the format and discussed solvability of implicit difference scheme.

Format construction

In order to obtain the numerical solution of the above equation, we introduce the general mesh generation:

, , , 1 , 0 , ), ,

(xj tk xj  jh j  M t_kk,k0,1,,N

where M,N are positive integer, hL/M is spatial orientation step,



T/N is time orientation step.

Let

k j

u

denote the exact solution in

(

x

j

,

t

k

)

_{point and}

U

kj _{denote the difference solution of this point.}

By the G-L formula, we obtain:

)

(

)

(

1

)

(

D

] / 1 [

0 1 1

0

_









O

k

t

f

t

f

k l y y

t









  

(10)

Where

 

,

0

,

1

,



)

1

(

_

1

_



y

_l

l l l



₍₁₁₎

We use compact operator   



  2 2

2

12 1

1 x

x

h 



to approach 2 2_u

x  

and then we get compact difference scheme for (6)-(9):

 













 ₂, 

h _.

Due to



0



1

_{, in the network point}



x

j

,

t

k



,

j



1

,

2

,



,

M



1

,

k



0

_{, we get}

  

   

 

 



 

 

   

  _{ }

 







  



N k

t U t U

M j

x U

f kU U h

k U

U

k k M k j

j j

k

l

k j l k j l k j

x x l

k j k j

, , 2 , 1 ), ( ),

(

, , , 2 , 1 , 0 ),

(

, ]

12 1 1 [

0 0

0 2 2

2 1

1

  

 

 

  

Let

 

  

  ₂, 

h _{. In the network point}

 

xj,tk ,j1,2,,M1,k0,1,,N_{, we get}

k j k

j k

j U U

U 1 ) 1

12 12 1 ( ) 6 5 2 6 5 ( ) 12 12 1 (                     

11 1

1 ₁₂)

12 1 ( k j U    









_













_



  



k j k j k j l j k l l

k

U

1

f

1

f

f

1

2 0

12

1

6

5

12

1

)

12

(









(13) Format analysis

Uniqueness of the numerical solution

Let

 





T k M k k

k _{ut u u}

u   1,, 1 _{be the exact solution vector. The matrix form of difference solution vector}

 



_k



T

M k k

k _{U t U U}

U   ₁,, _{1 is}

         



  1 0 1 0 ' 0 1 , , 3 , 2 , k l k l l

k _{BU F k N}

AU F U B AU  (14)

where l0,1,,k2,

k



2

,

' 0 1

B

B

_k



_,

                                           6 5 2 6 5 12 12

1 12 12

1 6 5 2 6 5 12 12 1 12 12 1 6 5 2 6 5 12 12

1 12 12

1 6 5 2 6 5                        A                                            6 5 2 6 5 12 12

1 12 12

1 6 5 2 6 5 12 12 1 12 12 1 6 5 2 6 5 12 12

1 12 12

1 6 5 2 6 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ' 0                                            B                                       6 5 2 12 12 6 5 2 12 12 6 5 2 12 12 6 5 2 1                    

k   

l

B

And A,B,Bl

'

0 are M1  M1 matrices. _M₁ Dimensional column vector can be expressed as

Let   



 _{ }

 _fk _fk _fk

P _{0 1 2}

12 1 6 5 12 1  ,      _{ }

 fMk fMk fMk Q 12 1 6 5 12 1 1 2 

, we get

                                   _{ }        _{ }               _{ }       _{ }        _{ }        _{ }         





          Q U U U f f f f f f P U U U F k M k M k l l M k k M k M k M k k k k k k l l k k 12 12 1 12 12 1 12 12 1 6 5 12 1 12 1 6 5 12 1 12 12 1 12 12 1 12 1 1 2 0 1 1 2 3 3 2 1 0 1 0 1 1 2 0 0 1                   

Theorem 3.1 Difference scheme (14) has the unique solution.

Proof: Obviously, constant matrix A is strictly diagonally dominant matrix for any 2 0 1    h K   

. Then A is nonsingular. So the solution of this compact format is being and unique [4-8].

Local truncation error:In the formula (10), let tk, f(t)1, we get ( ) ( ) 1 ( )

) ( 0 1 1 0 1 1     

  t _D   _O

k k

l l

t  

   



     .

So for any

t



T

, the local truncation error of (12) is



                          k l k j l k j t k j k j k j k

j _x u

u t x u t x u x D t u t u u R 0 1 2 2 1 2 2 1 0 1 ) ( )] , ( ) , ( [



_















   











k l l k j x x k j l

u

h

t

x

u

x

0 2 2

2 1 2 2 1

₎

)

12

1

(

)

,

(

(











 

)

(

)

(

)

240

(

)

(

4 0 2 6 1

h

O

O

h

u

O

k l k j x

l















 

_



_







 



.

Theoretical analysis: We use Fourier method to discuss the stability of difference scheme. Let

k j

U

'

be the approximate solution of (12), U Ujk j M k N

k j k

j   ,1  1,0 

'

 _{and the corresponding}

vector





T k M k k k 1 2

1, , 











 _{. Then we get}

k j k

j k

j 1 1

12 12 1 6 5 2 6 5 12 12 1         _{ }        _{ }        _

_

_



_

_



_

_



_

1 1 1 1 1 1 1 1 1 1

1 _{12 12}

1 6 5 2 6 5 12 12 1            _{ }        _{ }        _{ }  k j k j k j               

2 2 2

1 1 1 1 1

0 0 0

5 2

12 6 12

k k k

k j k j k j

l l l

























             _   _   _  _  _  _  _  _  _      







(15)

where j1,2,,M1,k1,2,N. Let

jh i k k j

d

e







and put it into (15). We get For

k



1

:

1 2 2 2 2 sin 3 2 sin 4 2 sin 3 1

1 h h hd

     _  _{ }  _ 

_

_

0 2 2 2 sin ) 1 ( 3 ) 1 ( 2 sin 1

4 h h_d



 _{    }

        

k

d

h

h

h













_

_

_

_

2

sin

3

2

sin

4

2

sin

3

1

1

2



_

2



_



2







1

2 2

2

sin

)

1

(

3

)

1

(

2

sin

1

4

_



_





_

_

_

_

_



d

k

h

h

_

_



_













2

2 2

1 0

4 1 sin ( 1)sin

2 3 2

k

k l

l

h h

d



















 

 

 _     _

 



(16)

In order to prove the stability of format, we introduce the following lemma.

Lemma 2.1[4] The constant



l



l



0

,

1

,





_satisfies

(1)



0



1

,



1







1

,



l



0

,

l



1

,

2

,



(2)









0

, 0

l l



for all





   n

l l n

1

1 ,

1 

Lemma 2.2 Assume dk(1kN) satisfies (16). So for 0



1, we get dk d0,k1,2,N_.

Proof: We use mathematical induction to prove. When

k



1

, we have

0 0 2 2

2

2 2

1

2 sin 3 1 1 2 sin 4 2 sin 3 1 1

2 sin 1 ) 1 ( 2 sin 4

d d h h

h

h h

d 

   

    



   

    

 _

   

 

  

Assume we have dn d0,1nk1_{. So for}

n



k

_{, from (15) we can get}





0 2 2

2

2 2

2 sin 3 2 sin 4 2 sin 3 1 1

2 sin 1 3 ) 1 ( 2 sin 4

d h h

h

h h

dk _{ }

   

 

    

  



   







0

1

0

1 2

2 2

2 2

2 sin 3 2

sin 4 2 sin 3 1 1

2 sin 1 3 2

sin ) 1 ( 4

d h

h h

h h

k

l l

k 

  



 _

  



   







 



































0 2 2

2

2 2

2 sin 3 2 sin 4 2 sin 3 1 1

2 sin 1 3 ) 1 2 sin ) 1 ( 4

d h h

h

h h

     

 

     

  



    















0 0

2 2

2

2 2

1 1

2 sin 3 2

sin 4 2 sin 3 1 1

2 sin 1 3 2

sin ) 1 ( 4

d d h

h h

h h

 

 

 



   



_

_

























Theorem 3.2 Difference scheme (12) is unconditionally stable for

0







1

. Proof: From Lemma 2.2 and Parsifal’s inequality, we can get









 

 



 

 

 1

1 2 1

1

2 1

1 2 2

2 '

2 2

M

j k M

j

jh i k M

j k j l

k l k

k _{U h h d e h dd}

U

_

_



N k

U U e

d h d h

l l

M

j jh i M

j

, , 2 , 1 ,

2 0 ' 0 2 0 2 1

1 0 1

1 2

0   2   2  









 







.

So we obtain the stability of (12). From local truncation error, we get that differential format is compatible with the original equation. According to Lax compatibility theorem and the proof of stability, we obtain theorem 3.3. Theorem 3.3 Difference scheme (12) is convergent.

CONCLUSION

Acknowledgement

We thank anonymous reviewers for helpful comments. This research is partially supported by the National Natural Science Foundation of China (No. 61170317) and the National Natural Science Foundation of Hebei Province (No. E2013209215).

REFERENCES

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